Chapter 17: Physical Properties of Gaussian Beams
=================================================

Source: Anthony E. Siegman, *Lasers* (1986), Chapter 17.
Use each section/problem identifier with the book; the original prompts are
not reproduced here.  Each entry gives the governing model, the decisive
solution route, and a physical verification.

Section 17.1: Gaussian Beam Propagation
---------------------------------------

Problem 17.1.1 — Gaussian beam transmission through a square aperture
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 17.1.2 — Criteria for centering accuracy of a circular aperture
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 17.1.3 — Setting tolerances on beam collimation and far-field beam angle
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 17.1.4 — Simulating an annular beam with positive and negative Gaussians
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 17.1.5 — Locating the center of curvature of a Gaussian beam
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 17.1.6 — Beam spot size at long distances
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Section 17.2: Gaussian Beam Focusing
------------------------------------

Problem 17.2.1 — Focusing for absolute minimum spot size
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Translate each performance requirement into an equality or inequality, solve the coupled constraints, and discard any root that violates a physical bound.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 17.2.2 — Focusing a Gaussian beam with an astigmatic lens
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 17.2.3 — Focusing into a dielectric medium
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Section 17.3: Lens Laws And Gaussian Mode Matching
--------------------------------------------------

Problem 17.3.1 — Thin-lens imaging formulas for Gaussian beams
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Section 17.5: Higher-Order Gaussian Modes
-----------------------------------------

Problem 17.5.1 — Instantaneous profile of the "donut" mode
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 17.5.2 — Near-field beam with a central hole
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Problem 17.5.3 — Recursion relation for expanding a half-rectangle in Hermite-Gaussian functions
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.

Section 17.6: Multimode Optical Beams
-------------------------------------

Problem 17.6.1 — Beam expansion criteria using Laguerre-Gaussian functions
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

List the supplied quantities in one unit system, isolate the requested variable symbolically, and retain guard digits until the final evaluation.
Represent the beam by :math:`q^{-1}=R^{-1}-j\lambda/(\pi w^2)` and propagate it with :math:`q_2=(Aq_1+B)/(Cq_1+D)`; separate real and imaginary parts only at the end.
Check that free propagation reproduces the Rayleigh-range formulas and that every computed spot size is real and positive.